I can't speak for the entire UK, but on the ECML (in GTR land) you'd generally look out for Arsenal fans, given the locality of the Emirates stadium.. and for Tottenham, Spurs fans. Obviously there are other fans going to other stadiums, and for longer distances you have the Intercity trains that could be packed out, with people more likely to drink (longer travel times) etc.
It's quite a good idea to avoid 2 hours before kick off and at least 2 hours after the game, adding on the usual stoppage time that Arsenal gets (so let's say 3 hours).
At Finsbury Park, it's just a case of shovelling people onto the train and moving it on. There are staff, but not enough if there are problems. Usual issue is just that people hold the doors open, or that the train is too full to close the doors, but - touch wood - never seen any fighting.
Problems have always been late at night, and when it's quieter as there's both room to fight and also less people to intervene (although that usually happens, or at least other passengers try and calm things down).
Once at night a fight broke out that ended up with a plain clothed officer on the train stepping in, being assaulted and the police coming en masse - complete with dogs. Not football related either.